over a network consisting of \(n\)  agents, where \(f_i(x):\mathbb {R}^m \rightarrow \mathbb {R}\)  is differentiable and only known by agent \(i\) . It is assumed that cenp has at least an optimal solution. The communication topology among agents is modeled by a digraph \(\mathcal {G}=(\mathcal {V}, \mathcal {E})\) , where \(\mathcal {V} = \lbrace 1,2,\cdots , n\rbrace \)  is the vertex set, \(\mathcal {E} \subset \mathcal {V} \times \mathcal {V}\)  is the edge set, and \((i, j) \in \mathcal {E}\)  if agents \(i\)  can receive information from agent \(j\) . Due to its promising applications on many areas, such as machine learning, distributed control and sensor networks [1]}, cenp has drawn lots of research interests and varieties of distributed algorithms have been proposed in the past decades, to name a few, distributed subgradient method [2]}, EXTRA [3]}, DLM [4]}, NIDS [5]}, Exact diffusion [6]}, and numerous gradient tracking methods [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, includes DIGing [7]}, [8]}, ATC-Tracking [9]}, [10]}, and Aug-DGM [11]}, [12]}. In this work, we aim to develop a general algorithmic framework which can unify existing algorithms. Some efforts have been made towards that goal, for example, [19]} unifies DIGing and EXTRA, [20]} unifies Exact diffusion and NIDS, and the two primal-dual frameworks proposed in [21]} and [22]} can unify most existing algorithms. However, the two primal-dual frameworks, which are the most general frameworks as far as we know, both rely on the symmetry of weight matrices, which is not necessary for many algorithms [8]}, [7]}, [9]}, [10]}, [11]}, [12]}, [6]}.
